For the first twenty-four hours, they did nothing except observe with passive instruments – telescopes, cameras, sensors on every wavelength. Vasili Orlov also took the opportunity of measuring the slab's dimensions with the greatest possible precision, and confirmed the famous 1:4:9 ratio to six decimal places. Big Brother was exactly the same shape as TMA-1 – but as it was more than two kilometres long, it was 718 times larger than its small sibling.

And there was a second mathematical mystery. Men had been arguing for years over that 1:4:9 ratio – the squares of the first three integers. That could not possibly be a coincidence; now here was another number to conjure with.

Back on Earth, statisticians and mathematical physicists were soon playing happily with their computers, trying to relate the ratio to the fundamental constants of nature – the velocity of light, the proton/electron mass ratio, the fine-structure constant. They were quickly joined by a gaggle of numerologists, astrologers, and mystics, who threw in the height of the Great Pyramid, the diameter of Stonehenge, the azimuth bearings of the Nazca lines, the latitude of Easter Island, and a host of other factors from which they were able to draw the most amazing conclusions about the future. They were not in the least deterred when a celebrated Washington humorist claimed that his calculations proved that the world ended on 31 December 1999 – but that everyone had had too much of a hangover to notice.

Nor did Big Brother appear to notice the two ships that had arrived in its vicinity – even when they cautiously probed it with radar beams and bombarded it with strings of radio pulses which, it was hoped, would encourage any intelligent listener to answer in the same fashion.

After two frustrating days, with the approval of Mission Control, the ships halved their distance. From fifty kilometres, the largest face of the slab appeared about four times the width of the Moon in Earth's sky – impressive, but not so large as to be psychologically overwhelming. It could not yet compete with Jupiter, ten times larger still; and already the mood of the expedition was changing from awed alertness to a certain impatience.

Walter Curnow spoke for almost everyone: 'Big Brother may be willing to wait a few million years – we'd like to get away a little sooner.'

Discovery had left Earth with three of the little space pods that allowed an astronaut to perform extravehicular activities in shirt-sleeve comfort. One had been lost in the accident – if it was an accident – that had killed Frank Poole. Another had carried Dave Bowman to his final appointment with Big Brother, and shared whatever fate befell him, A third was still in the ship's garage, the Pod Bay.

It lacked one important component – the hatch, blown off by Commander Bowman when he had made his hazardous vacuum-crossing and entered the ship through the emergency airlock, after Hal had refused to open the Pod Bay door. The resulting blast of air had rocketed the pod several hundred kilometres away before Bowman, busy with more important matters, had brought it back under radio control. It was not surprising that he had never bothered to replace the missing hatch.

Now Pod Number 3 (on which Max, refusing all explanations, had stencilled the name Nina) was being prepared for another EVA. It still lacked a hatch, but that was unimportant. No one would be riding inside.

Bowman's devotion to duty was a piece of unexpected luck, and it would have been folly not to take advantage of it. By using Nina as a robot probe, Big Brother could be examined at close quarters without risking human lives. That at least was the theory; no one could rule out the possibility of a backlash that might engulf the ship. After all, fifty kilometres was not even a hair's breadth, as cosmic distances went.

After years of neglect, Nina looked distinctly shabby. The dust that was always floating around in zero gee had settled over the outer surface, so that the once immaculately white hull had become a dingy grey. As it slowly accelerated away from the ship, its external manipulators folded neatly back and its oval viewport staring spaceward like a huge, dead eye, it did not seem a very impressive ambassador of Mankind. But that was a distinct advantage; so humble an emissary might be tolerated, and its small size and low velocity should emphasize its peaceful intentions. There had been a suggestion that it should approach Big Brother with open hands; the idea was quickly turned down when almost everyone agreed that if they saw Nina heading toward them, mechanical claws outstretched, they would run for their lives.

After a leisurely two-hour trip, Nina came to rest a hundred metres from one corner of the huge rectangular slab. From so close at hand, there was no sense of its true shape; the TV cameras might have been looking down on the tip of a black tetrahedron of indefinite size. The onboard instruments showed no sign of radioactivity or magnetic fields; nothing whatsoever was coming from Big Brother except the tiny fraction of sunlight it condescended to reflect.

After five minutes' pause – the equivalent, it was intended, of 'Hello, here I am!' – Nina started a diagonal crossing of the smaller face, then the next larger, and finally the largest, keeping at a distance of about fifty metres, but occasionally coming in to five. Whatever the separation, Big Brother looked exactly the same – smooth and featureless. Long before the mission was completed, it had become boring, and the spectators on both ships had gone back to their various jobs, only glancing at the monitors from time to time.

'That's it,' said Walter Curnow at last, when Nina had arrived back where she had started. 'We could spend the rest of our lives doing this, without learning anything more. What do I do with Nina – bring her home?'

'No,' said Vasili, breaking into the circuit from aboard Leonov. 'I've a suggestion. Take her to the exact centre of the big face. Bring her to rest – oh, a hundred metres away. And leave her parked there, with the radar switched to maximum precision.'

'No problem – except that there's bound to be some residual drift. But what's the point?'

'I've just remembered an exercise from one of my college astronomy courses – the gravitational attraction of an infinite flat plate. I never thought I'd have a chance of using it in real life. After I've studied Nina's movements for a few hours, at least I'll be able to calculate Zagadka's mass, That is, if it has any. I'm beginning to think there's nothing really there.'

'There's an easy way to settle that, and we'll have to do it eventually. Nina must go in and touch the thing.'

'She already has.'

'What do you mean?' asked Curnow, rather indignantly. 'I never got nearer than five metres.'

'I'm not criticizing your driving skills – though it was a pretty close thing at that first corner, wasn't it? But you've been tapping gently on Zagadka every time you use Nina's thrusters near its surface.'

'A flea jumping on an elephant!'

'Perhaps. We simply don't know. But we'd better assume that, one way or another, it's aware of our presence, and will only tolerate us as long as we aren't a nuisance.'

He left the unspoken question hanging in the air. How did one annoy a two-kilometre-long black rectangular slab? And just what form would its disapproval take?